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Two circles of radius 4 are externally tangent to each other and are internally tangent to a circle of radius 10 at points A and B.  Find the distance AB.

 

 Jun 1, 2021
 #1
avatar+1776 
+1

If we draw the centers for all the circles and connect them, we get a triangle of 8, 8, 8. 

If we draw a triangle for the center of the large triangle and point A and B, we have a similar triangle with 2 sides being 10, meaning the side AB is also 10. 

 

=^._.^=

 Jun 1, 2021
 #3
avatar+1776 
+1

Edit:

CPhill is right, the triangle isn't 8, 8, 8.

Oops.

 

=^._.^=

catmg  Jun 1, 2021
 #2
avatar+119780 
+2

I think  we  can use  the idea of similar  triangles  here

 

Call  the center of  the large circle O   and  the  centers   of   the  smaller  circles =  C  and D

 

OC  =  (10 - 4)  = 6     CD  = (4 + 4)  =   8

OA =  10     AB =  ???

 

And triangles   OCD    and OAB  are similar

 

So

 

CD / OC  =   AB / OA

 

8 / 6  =  AB / 10

 

10 * 8 / 6   =  AB 

 

10   ( 4/3)   = AB

 

40/3  =  AB =  13   1/3 

 

 

cool cool cool

 Jun 1, 2021

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